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Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $s \geq 0$ be an integer, and let $T_{2 π}^{3}$ , $μ$ , $χ_{k}$ , $\hat{u} (k)$ , the weak derivatives $D^{α}$ and the spaces $H^{s} (T_{2 π}^{3}; R^{N})$ be as in The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s, and put $N_{s} (u)^{2} : = \sum_{k \in Z^{3}} (1 + ∣ k ∣^{2})^{s} ∣ \hat{u} (k) ∣^{2}, ∣ k ∣^{2} : = k_{1}^{2} + k_{2}^{2} + k_{3}^{2} .$

(a) Let $u \in H^{s}$ and let $α \in N_{0}^{3}$ with $∣ α ∣ \leq s$ . Then $D^{α} u \in H^{s - ∣ α ∣}$ and $\hat{D^{α} u} (k) = (i k)^{α} \hat{u} (k) (k \in Z^{3}), (i k)^{α} : = \prod_{j = 1}^{3} (i k_{j})^{α_{j}} .$

(b) Let $u \in L^{2} (T_{2 π}^{3}; R^{N})$ satisfy $N_{s} (u) < \infty$ . Then $u \in H^{s} (T_{2 π}^{3}; R^{N})$ , and there are constants $0 < c_{s} \leq C_{s} < \infty$ , depending only on $s$ , such that $c_{s} N_{s} (u)^{2} \leq ∥ u ∥_{H^{s}}^{2} \leq C_{s} N_{s} (u)^{2} .$ The right-hand inequality holds for every $u \in H^{s}$ , so the two norms are equivalent on $H^{s}$ .

Facts & Assumptions

Given: Countable Choice, an integer $s \geq 0$ , the torus $T_{2 π}^{3}$ with its normalized measure $μ$ and characters $χ_{k}$ , the weak derivatives $D^{α}$ , and the spaces $H^{s}$ .

[L1]

$(χ_{k})_{k \in Z^{3}}$ is an orthonormal basis of the complex $L^{2}$ : for $f, g \in L^{2}$ one has the norm and pairing identities $∥ f ∥_{2}^{2} = \sum_{k} ∣ \hat{f} (k) ∣^{2}$ and $⟨ f, g ⟩ = \sum_{k} \hat{f} (k) \overline{\hat{g} (k)}$ , the series converging absolutely, and the coefficient map is a surjective linear isometry onto $ℓ^{2} (Z^{3})$ (The Fourier basis and Parseval's identity on the finite torus, The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s).

[L2]

$v = D^{α} u$ means $\int u D^{α} φ d μ = (- 1)^{∣ α ∣} \int v φ d μ$ for every smooth test function $φ$ ; the real and imaginary parts of $χ_{l}$ are smooth real tests; adding their test identities extends the identity complex-linearly to $χ_{l}$ (The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s).

[L3]

For a $C^{1}$ function $G$ on an interval $[a, b]$ , $\int_{a}^{b} G^{'} = G (b) - G (a)$ : the derivative $G^{'}$ is continuous, hence integrable, and the second fundamental theorem of calculus applies (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

[L4]

Multinomial expansion: for any finite real tuple $(x_{1}, \dots, x_{d})$ and $s \in N_{0}$ , $(x_{1} + \dots + x_{d})^{s} = \sum_{∣ γ ∣ = s} (s! / γ!) x^{γ}$ , over the finite set of $d$ -component multi-indices. The coefficients are nonnegative integers and sum to $d^{s}$ , by taking all $x_{i} = 1$ (The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Proof

technique · direct

1.1L2L3algebra

(One-variable periodic integration by parts.) For smooth $2 π$ -periodic $G, H$ and each $j$ , $\int_{T_{2 π}} G \partial_{j} H d μ_{T} = - \int_{T_{2 π}} (\partial_{j} G) H d μ_{T}$ : indeed $\int_{0}^{2 π} \partial_{j} (G H) d x_{j} = (G H) ∣_{0}^{2 π} = 0$ by the second fundamental theorem of calculus and periodicity [L3], and the product rule gives the claim; the remaining coordinates integrate out by the product-measure convention. Consequently, for every smooth $φ$ , every $k$ and every $α$ , $\int χ_{k} D^{α} φ d μ = (- 1)^{∣ α ∣} (i k)^{α} \int χ_{k} φ d μ$ : induct on $∣ α ∣$ , moving one variable derivative from $φ$ onto $χ_{k}$ by the one-variable identity and using $\partial_{j} χ_{k} = i k_{j} χ_{k}$ . In particular $D^{α} χ_{k} = (i k)^{α} χ_{k}$ in the weak sense of [L2], the case $φ = χ_{- l}$ tested against $χ_{k}$ having Fourier coefficients $\hat{χ_{k}} (l) = δ_{k l}$ .

1.2L4algebra

(The symbol bounds.) For $k \in Z^{3}$ put $q_{k} : = \sum_{∣ α ∣ \leq s} ∣ k^{α} ∣^{2}$ . Each term is at most $(1 + ∣ k ∣^{2})^{s}$ , and there are at most $(s + 1)^{3}$ terms, so $q_{k} \leq (s + 1)^{3} (1 + ∣ k ∣^{2})^{s}$ . Conversely, the multinomial expansion of $(1 + k_{1}^{2} + k_{2}^{2} + k_{3}^{2})^{s}$ is $\sum_{∣ α ∣ \leq s} c_{α} ∣ k^{α} ∣^{2}$ , where $c_{α} = s! / ((s - ∣ α ∣)! α!)$ . These nonnegative integer coefficients sum to $4^{s}$ by setting all four variables to $1$ , so every $c_{α} \leq 4^{s}$ . It follows that $(1 + ∣ k ∣^{2})^{s} \leq 4^{s} q_{k}$ . This includes $s = 0$ , where both sides are $1$ . Hence $4^{- s} (1 + ∣ k ∣^{2})^{s} \leq q_{k} \leq (s + 1)^{3} (1 + ∣ k ∣^{2})^{s}$ .

2.1step 1.1L2algebra

(The differentiation identity (a).) If $u \in H^{s}$ and $∣ α ∣ \leq s$ , then $D^{α} u \in H^{s - ∣ α ∣}$ : a weak derivative of $D^{α} u$ of order $∣ β ∣ \leq s - ∣ α ∣$ is the weak derivative $D^{α + β} u$ , because the defining test identity for the iterated derivative follows by applying the two defining identities in succession and noting that $D^{β} φ$ is again a smooth test function. Testing the defining identity for $D^{α} u$ against $φ = χ_{- k}$ gives $\hat{D^{α} u} (k) = (- 1)^{∣ α ∣} \int u D^{α} χ_{- k} d μ = (- 1)^{∣ α ∣} (- i k)^{α} \int u χ_{- k} d μ = (i k)^{α} \hat{u} (k)$ , where the middle equality uses step 1.1 for the smooth test function $D^{α} χ_{- k} = (- i k)^{α} χ_{- k}$ and the last uses $(- 1)^{∣ α ∣} (- i)^{α} = i^{α}$ .

2.2step 1.2L1algebra

(The norm bounds for $u \in H^{s}$ .) By Parseval [L1] applied to the $L^{2}$ classes $D^{α} u$ , $∥ D^{α} u ∥_{2}^{2} = \sum_{k} ∣ k^{α} ∣^{2} ∣ \hat{u} (k) ∣^{2}$ , so the definition of the $H^{s}$ norm gives $∥ u ∥_{H^{s}}^{2} = \sum_{k} q_{k} ∣ \hat{u} (k) ∣^{2}$ . Combining with step 1.2, $∥ u ∥_{H^{s}}^{2} \leq (s + 1)^{3} N_{s} (u)^{2}$ and $N_{s} (u)^{2} \leq 4^{s} ∥ u ∥_{H^{s}}^{2}$ , which are the right-hand inequality and the converse inequality of (b) whenever $u \in H^{s}$ .

3.1step 2.1L1L2algebra

(Converse direction of (b): the weighted series produce weak derivatives.) Let $u \in L^{2}$ with $N_{s} (u) < \infty$ . For each $∣ α ∣ \leq s$ the sequence $k \mapsto (i k)^{α} \hat{u} (k)$ is square-summable because $∣ k^{α} ∣^{2} \leq (1 + ∣ k ∣^{2})^{∣ α ∣} \leq (1 + ∣ k ∣^{2})^{s}$ ; by the surjectivity of the coefficient isometry [L1] there is a unique class $w_{α} \in L^{2}$ with $\hat{w_{α}} (k) = (i k)^{α} \hat{u} (k)$ . Since $u$ is real, $\hat{u} (- k) = \overline{\hat{u} (k)}$ by conjugating its integral. Thus $a_{k} : = (i k)^{α} \hat{u} (k)$ satisfies $a_{- k} = \overline{a_{k}}$ . Also $\hat{\overline{w_{α}}} (k) = \overline{\hat{w_{α}} (- k)} = a_{k}$ , again by conjugating the integral. Injectivity in [L1] gives $\overline{w_{α}} = w_{α}$ as $L^{2}$ classes, so $w_{α}$ has a real-valued representative, componentwise. For a smooth real test $φ$ , Parseval's pairing identity [L1] gives $\int u D^{α} φ d μ = \sum_{k} \hat{u} (k) \overline{\hat{D^{α} φ} (k)} = \sum_{k} \hat{u} (k) \overline{(i k)^{α} \hat{φ} (k)} = \sum_{k} (- 1)^{∣ α ∣} (i k)^{α} \hat{u} (k) \overline{\hat{φ} (k)} = (- 1)^{∣ α ∣} \int w_{α} φ d μ$ , where the second equality is step 2.1 applied to the smooth $φ$ and the third uses $\overline{(i k)^{α}} = (- i)^{α} k^{α} = (- 1)^{∣ α ∣} (i k)^{α}$ . Hence $D^{α} u = w_{α}$ weakly for every $∣ α ∣ \leq s$ , so $u \in H^{s}$ .

4.1step 3.1step 2.1step 2.2constructdischarge-construct∎

(Conclusion.) By step 3.1 and step 2.2, for $u$ with $N_{s} (u) < \infty$ the class lies in $H^{s}$ and satisfies $4^{- s} N_{s} (u)^{2} \leq ∥ u ∥_{H^{s}}^{2} \leq (s + 1)^{3} N_{s} (u)^{2}$ , so (b) holds with $c_{s} = 4^{- s}$ and $C_{s} = (s + 1)^{3}$ , while (a) was proved in step 2.1.

Remarks

Use. Part (a) is the differentiation rule and (b) the equivalence of norms used freely on this page: Periodic Sobolev embedding: H^s on the three-torus controls C^m when s minus m exceeds three halves uses (b) to reduce $L^{\infty}$ bounds to weighted coefficient sums, and Fourier modes on the periodic three-torus and the Galerkin projection P_N uses (a) to commute the projection with derivatives.
Constants. No sharpness of $c_{s}, C_{s}$ is claimed; only their existence and their dependence on $s$ alone.

Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources